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Bounds for the least solutions of homogeneous quadratic equations

Published online by Cambridge University Press:  24 October 2008

J. W. S. Cassels
J. W. S. Cassels
Affiliation:
Trinity CollegeCambridge
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In this note I obtain bounds for the least integral solutions of the equation

in terms of

For ternary diagonal forms, such bounds have been given by Axel Thue (4) and, more recently, by Holzer (1), Mordell (2) and Skolem (3), but these lead only to bad estimates for general ternaries. So far as I know there have not been given estimates for n≽4. Here I generalize Thue's method to prove:

Theorem. Suppose that n ≥ 2 and that f(ɛ) represents zero. Then there is an integral solution of f(a) = 0 with

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 51 , Issue 2 , April 1955 , pp. 262 - 264
Copyright
Copyright © Cambridge Philosophical Society 1955

References

REFERENCES

(1)Holzer, L.Minimal solutions of Diophantine equations. Canad. J. Math. 2 (1950), 238–44.CrossRefGoogle Scholar
(2)Mordell, L. J.On the equation ax 2 + by 2 = cz 2. Mh. Math. 55 (1951), 323–7.CrossRefGoogle Scholar
(3)Skolem, T.Et enkelt bevis for løsbarhetsbetingelsen for den diofantiske ligning ax 2 + by 2 + cz 2 = 0. Norsk mat. Tidsskr. 33 (1951), 105–12.Google Scholar
(4)Thue, A.Eine Eigenschaft der Zahlen der Fermat'schen Gleichung. Skr. VidenskSelsk., Christ., (Mat.-naturv. Kl.), 1911 3.Google Scholar